Nonlinear Q-difference Equations Research Articles

On existence of meromorphic solutions for nonlinear q-difference equation

On existence of meromorphic solutions for nonlinear q-difference equation

Power Series Solutions of Non-linear q-Difference Equations and the Newton–Puiseux Polygon

Adapting the Newton–Puiseux Polygon process to nonlinear q-difference equations of any order and degree, we compute their power series solutions, study the properties of the set of exponents of the solutions and give a bound for their q-Gevrey order in terms of the order of the original equation.

On a nonlinear sequential four-point fractional q-difference equation involving q-integral operators in boundary conditions along with stability criteria

In this paper, we consider a nonlinear sequential q-difference equation based on the Caputo fractional quantum derivatives with nonlocal boundary value conditions containing Riemann–Liouville fractional quantum integrals in four points. In this direction, we derive some criteria and conditions of the existence and uniqueness of solutions to a given Caputo fractional q-difference boundary value problem. Some pure techniques based on condensing operators and Sadovskii’s measure and the eigenvalue of an operator are employed to prove the main results. Also, the Ulam–Hyers stability and generalized Ulam–Hyers stability are investigated. We examine our results by providing two illustrative examples.

Behaviour of the First-order q-Dierence Equations

Since the need to investigate many aspects of q-dierence equations cannot be ruled out, this article aims to explore response of the mechanism modelled by linear and nonlinear q-difference equations. Therefore, analysis of an important bundle of nonlinear q-difference equations, in particular the q-Bernoulli difference equation, has been developed. In this context, capturing the behaviour of the q-Bernoulli difference equation as well as linear q-difference equations are considered to be a significant contribution here. Illustrative examples related to the difference equations are also presented.

A q-analogue of the matrix sixth Painlevé system

We derive a q-analogue of the matrix sixth Painlevé system from a connection-preserving deformation of a certain Fuchsian linear q-difference system. In specifying the linear q-difference system, we utilize the correspondence between linear differential systems and linear q-difference systems from the viewpoint of the spectral type. The system of non-linear q-difference equations thus obtained can also be regarded as a non-abelian analogue of Jimbo–Sakai’s q–P VI.

Existence of positive strongly decaying solutions of second-order nonlinear q-difference equations

This paper investigates the existence of positive strongly decaying solutions of second-order nonlinear q-difference equation with q-regularly varying coefficients a and b. By means of a fixed point theorem in partially ordered Banach space, necessary and sufficient conditions for the existence of solutions satisfying and as are established. Moreover, with the help of q-regular varying theory, precise asymptotic behaviour of such solutions is described.

Analytic solutions of q-P(A1) near its critical points

For transcendental functions that solve non-linear q-difference equations, the best descriptions available are the ones obtained by expansion near critical points at the origin and infinity. We describe such solutions of a q-discrete Painlevé equation, with seven parameters whose initial value space is a rational surface of type . The resultant expansions are shown to approach series expansions of the classical sixth Painlevé equation in the continuum limit.

ON EXISTENCE OF SOLUTIONS FOR NONLINEAR Q-DIFFERENCE EQUATIONS WITH NONLOCAL Q-INTEGRAL BOUNDARY CONDITIONS

In this paper, we discuss the existence of solutions for nonlinear qdifference equations with nonlocal q-integral boundary conditions. The first part of the paper deals with some existence and uniqueness results obtained by means of standard tools of fixed point theory. In the second part, sufficient conditions for the existence of extremal solutions for the given problem are established. The results are well illustrated with the aid of examples.

Existence and growth of meromorphic solutions of some nonlinear q-difference equations

In this paper, we investigate the existence of transcendental meromorphic solutions of order zero of some nonlinear q-difference equations and some more general equations. We also give some estimates on the growth of transcendental meromorphic solutions of these equations. Some examples are given to illustrate the sharpness of some of our results.

Three-point boundary value problems for nonlinear q-difference equations with multi-term q-derivative boundary conditions

In this paper, we present some new existence and uniqueness results for three-point boundary value problems of nonlinear q-difference equations with multi-term q-derivative boundary conditions. Our results are based on Banach’s contraction principle, Krasnowselskii’s fixed point theorem and Leray-Schauder degree theory. Two examples are given to illustrate the advantage of our results.

Sequential Derivatives of Nonlinearq-Difference Equations with Three-Pointq-Integral Boundary Conditions

This paper studies sufficient conditions for the existence of solutions to the problem of sequential derivatives of nonlinearq-difference equations with three-pointq-integral boundary conditions. Our results are concerned with several quantum numbers of derivatives and integrals. By using Banach's contraction mapping, Krasnoselskii's fixed-point theorem, and Leray-Schauder degree theory, some new existence results are obtained. Two examples illustrate our results.

On nonlocal boundary value problems of nonlinear q-difference equations

This paper studies a nonlocal boundary value problem of nonlinear third-order q-difference equations. Our results are based on Leray-Schauder degree theory and some standard fixed point theorems.

Irreducibility of q-Painlevé equation of type A 6 (1) in the sense of order

In this paper we will study the irreducibility of q-, q-Painlevé equation of type , in the sense of order using the notion of decomposable extensions. q- is one of the special non-linear q-difference equations of order 2 with symmetry and is also called q-.

Variational Iteration Method for q‐Difference Equations of Second Order

Recently, Liu extended He′s variational iteration method to strongly nonlinear q‐difference equations. In this study, the iteration formula and the Lagrange multiplier are given in a more accurate way. The q‐oscillation equation of second order is approximately solved to show the new Lagrange multiplier′s validness.

Existence of analytic solutions to analytic nonlinear q-difference equations

In this paper, some analytic approaches are formulated for the existence of analytic solutions of analytic nonlinear difference equations. From the point of view of dynamical systems, analytic solutions of such kinds of equations can be generally expressed by formal power series of exponential variables, so we are interested in considering a difference equation as a q-difference equation via a suitable coordinate transformation. After stating analytic results for formal series solutions of nonlinear q-difference equations, we also derive some results for the existence of analytic solutions to autonomous rational difference equations.

An ultrametric version of the Maillet-Malgrange theorem for nonlinear $q$-difference equations

We prove an ultrametric q-difference version of the Maillet-Malgrange theorem on the Gevrey nature of formal solutions of nonlinear analytic q-difference equations. Since deg q and ord q define two valuations on C(q), we obtain, in particular, a result on the growth of the degree in q and the order at q of formal solutions of nonlinear q-difference equations, when q is a parameter. We illustrate the main theorem by considering two examples: a q-deformation of Painleve II, for the nonlinear situation, and a q-difference equation satisfied by the colored Jones polynomials of the figure 8 knots, in the linear case. We also consider a q-analog of the Maillet-Malgrange theorem, both in the complex and in the ultrametric setting, under the assumption that |q| = 1 and a classical diophantine condition.

Analytic Solutions of aq-Difference Equation and Applications to Iterative Equations*

In this paper, a nonlinear q-difference equation is investigated in the complex field C for existence of local invertible analytic solutions. We discuss not only in the case but also for Our results can be applied to obtain existence of analytic solutions for some iterative equations.

Asymptotic Discontinuity of Smooth Solutions of Nonlinear q-Difference Equations

We investigate the asymptotic behavior of solutions of the simplest nonlinear q-difference equations having the form x(qt+ 1) = f(x(t)), q> 1, t∈ R+. The study is based on a comparison of these equations with the difference equations x(t+ 1) = f(x(t)), t∈ R+. It is shown that, for “not very large” q> 1, the solutions of the q-difference equation inherit the asymptotic properties of the solutions of the corresponding difference equation; in particular, we obtain an upper bound for the values of the parameter qfor which smooth bounded solutions that possess the property \(\begin{array}{*{20}c} {\max } \\ {t \in [0,T]} \\ \end{array} \left| {x'(t)} \right| \to \infty \)as T→ ∞ and tend to discontinuous upper-semicontinuous functions in the Hausdorff metric for graphs are typical of the q-difference equation.

Long-time properties of solutions of simplest nonlinear q-difference equations

We investigate the asymptotic behavior of solutions for simplest nonlinear q-difference equations of the form The basis for the study is a comparison between these equations (at various q) and the difference equation . We show, that for “ moderate ” q>1 solutions of the above q-difference equations inherit the long-time properties of solutions of the corresponding difference equation.